2.2 The limit of a function  (Page 6/12)

We can use [link] directly.

1. $\underset{x\to {\mathrm{-3}}^{-}}{\text{lim}}\frac{1}{{\left(x+3\right)}^4}=\text{+}\infty$
2. $\underset{x\to {\mathrm{-3}}^{+}}{\text{lim}}\frac{1}{{\left(x+3\right)}^4}=\text{+}\infty$
3. $\underset{x\to \mathrm{-3}}{\text{lim}}\frac{1}{{\left(x+3\right)}^4}=\text{+}\infty$

The function $f\left(x\right)=1\text{/}{\left(x+3\right)}^4$ has a vertical asymptote of $x=\mathrm{-3}.$

Using [link] and the graph for reference, we arrive at the following values:

1. $\underset{x\to {\mathrm{-4}}^{-}}{\text{lim}}f\left(x\right)=0;\underset{x\to {\mathrm{-4}}^{+}}{\text{lim}}f\left(x\right)=0;\underset{x\to \mathrm{-4}}{\text{lim}}f\left(x\right)=0;f\left(\mathrm{-4}\right)=0$
2. $\underset{x\to {\mathrm{-2}}^{-}}{\text{lim}}f\left(x\right)=3.;\underset{x\to {\mathrm{-2}}^{+}}{\text{lim}}f\left(x\right)=3;\underset{x\to \mathrm{-2}}{\text{lim}}f\left(x\right)=3;f\left(\mathrm{-2}\right)$ is undefined
3. $\underset{x\to 1^{-}}{\text{lim}}f\left(x\right)=6;\underset{x\to 1^{+}}{\text{lim}}f\left(x\right)=3;\underset{x\to 1}{\text{lim}}f\left(x\right)$ DNE; $f\left(1\right)=6$
4. $\underset{x\to 3^{-}}{\text{lim}}f\left(x\right)=\text{−}\infty ;\underset{x\to 3^{+}}{\text{lim}}f\left(x\right)=\text{−}\infty ;\underset{x\to 3}{\text{lim}}f\left(x\right)=\text{−}\infty ;f\left(3\right)$ is undefined

Our starting point is Einstein’s equation for the mass of a moving object,

where $m_0$ is the object’s mass at rest, v is its speed, and c is the speed of light. To see how the mass changes at high speeds, we can graph the ratio of masses $m\text{/}{m}_0$ as a function of the ratio of speeds, $v\text{/}c$ ( [link] ).

We can see that as the ratio of speeds approaches 1—that is, as the speed of the object approaches the speed of light—the ratio of masses increases without bound. In other words, the function has a vertical asymptote at $v\text{/}c=1.$ We can try a few values of this ratio to test this idea.

Thus, according to [link] , if an object with mass 100 kg is traveling at 0.9999 c , its mass becomes 7071 kg. Since no object can have an infinite mass, we conclude that no object can travel at or more than the speed of light.